why a
(-)x(-)=(+)
(-)x(+)= (-)
Those are the rules for multiplication of signed numbers.
If x>0 and a>0 and accepting that (+)(+) =(+) as well as the distributive property, lets investigate
x(a + (-a)), expanding we get
ax + (-a)x
we know that a + (-a) is zero, since the sum of two opposites is zero.
So the result above has to be zero.
Therefore (-a)x = -ax and a positive times a negative is negative.
now repeat the above steps with
(-x)(a + (-a)), again this result should be zero
expanding by the distributive property we get
(-x((a) + (-x)(-a)
we established that (-x)(a) is -ax, that is it is a negative number, so (-x)(-a) has to be its positive opposite to make the sum zero
so (-a)(-x) = +ax
I used to show the rules by relating them to playing a game in sports
Assume a good player is +
and a poor player is -
Assume playing is +
and sitting on the bench is -
1. A good player is playing, that's good +x+ = +
2. A poor player is playing, not good -x+ = -
3. A good player is on the bench, not good +x- = -
4. A poor player on the bench, that's good -x- = +
The rules for multiplying positive and negative numbers can be explained using the concept of multiplication as repeated addition.
When we multiply two positive numbers, such as 3 and 4, we're essentially adding 4, three times: 4 + 4 + 4 = 12. This makes intuitive sense since multiplication is often thought of as "repeated addition."
Now, let's consider the multiplication of a positive number and a negative number. If we take 3 multiplied by -4, we are essentially adding -4, three times: -4 + (-4) + (-4) = -12. In this case, the negative sign in front of the 4 indicates that each term we add is negative, leading to a negative product.
Similarly, when we multiply a negative number by a positive number, say -3 and 4, we are adding 4, three times with a negative sign in front: -(4 + 4 + 4) = -12. Here, the negative sign in front of the 3 indicates that each term we add is negative, resulting in a negative product.
Finally, let's consider multiplying two negative numbers, such as -3 and -4. Following the same logic, we would add -4, three times, with negative signs in front: -(-4 + (-4) + (-4)).
To simplify this expression, we can apply the rule that double negatives cancel each other out. So, -(-4) becomes +(4), resulting in 4 + 4 + 4 = 12. In this case, the two negative signs in front of the numbers cancel each other out, leading to a positive product.
So, the reason why multiplying two negative numbers results in a positive product is because the negative signs cancel each other out.
(-) x (-) = (+)
(-) x (+) = (-)